Egyptian mathematics is the
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
that was developed and used in
Ancient EgyptAncient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...
from ca. 3000 BC to ca. 300 BC.
Overview
Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found at Tomb Uj at
AbydosAbydos is one of the most ancient cities of Upper Egypt, and also of the eight Upper Nome, of which it was the capital city. It is located about 11 kilometres west of the Nile at latitude 26° 10' N, near the modern Egyptian towns of el'Araba el Madfuna and alBalyana...
. These labels appear to have been used as tags for grave goods and some are inscribed with numbers. Further evidence of the use of the base 10 number system can be found on for instance the
Narmer MaceheadThe Narmer macehead is an ancient Egyptian decorative stone mace head. It was found during a dig at Kom al Akhmar, the site of Hierakonpolis It is dated to the reign of king Narmer whose serekh is engraved on it...
which depicts offerings of 400,000 oxen, 1,422,000 goats and 120,000 prisoners.
The evidence of the use of mathematics in the
Old KingdomOld Kingdom is the name given to the period in the 3rd millennium BC when Egypt attained its first continuous peak of civilization in complexity and achievement – the first of three socalled "Kingdom" periods, which mark the high points of civilization in the lower Nile Valley .The term itself was...
(ca 2690  2180 BC) is scarce, but can be deduced from for instance inscriptions on a wall near a
mastabaA mastaba, or "prdjt" , is a type of ancient Egyptian tomb in the form of a flatroofed, rectangular structure with outward sloping sides that marked the burial site of many eminent Egyptians of Egypt's ancient period...
in
MeidumLocated about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mudbrick mastabas.Pyramid:...
which gives guidelines for the slope of the mastaba. The lines in the diagram are spaced at a distance of one
cubitThe cubit is a traditional unit of length, based on the length of the forearm. Cubits of various lengths were employed in many parts of the world in Antiquity, in the Middle Ages and into Early Modern Times....
and show the use of that
unit of measurementLength:Units of length date back to at least the early dynastic period. In the Palermo stone for instance the level of the Nile river is recorded. During the reign of Pharaoh Djer the height of the river Nile was given as measuring 6 cubits and 1 palm...
.
The earliest true mathematical documents date to the
12th dynastyThe twelfth dynasty of ancient Egypt is often combined with Dynasties XI, XIII and XIV under the group title Middle Kingdom.Rulers:Known rulers of the twelfth dynasty are as follows :...
(ca 1990  1800 BC). The
Moscow Mathematical PapyrusThe Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
, the
Egyptian Mathematical Leather RollThe Egyptian Mathematical Leather Roll was a 10 × 17 in leather roll purchased by Alexander Henry Rhind in 1858...
, the
Lahun Mathematical PapyriThe Lahun Mathematical Papyri are part of a collection of Kahun Papyri discovered at ElLahun by Flinders Petrie during excavations of a worker's town near the pyramid of Sesostris II...
which are a part of the much larger collection of Kahun Papyri and the
Berlin PapyrusThe Berlin Papyrus 6619, commonly known as the Berlin Papyrus, is an ancient Egyptian papyrus document from the Middle Kingdom. This papyrus was found at the ancient burial ground of Saqqara in the early 19th century CE....
all date to this period. The
Rhind Mathematical PapyrusThe Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
which dates to the Second Intermediate Period (ca 1650 BC) is said to be based on an older mathematical text from the 12th dynasty.
The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are socalled mathematical problem texts. They consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems.
An interesting feature of Ancient Egyptian mathematics is the use of unit fractions. The Egyptians used some special notation for fractions such as
and
and in some texts for
, but other fractions were all written as
unit fractions of the form
or sums of such unit fractions. Scribes used tables to help them work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of other unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain
tables. These tables allowed the scribes to rewrite any fraction of the form
as a sum of unit fractions.
During the
New KingdomThe New Kingdom of Egypt, also referred to as the Egyptian Empire is the period in ancient Egyptian history between the 16th century BC and the 11th century BC, covering the Eighteenth, Nineteenth, and Twentieth Dynasties of Egypt....
(ca 1550  1070 BC) mathematical problems are mentioned in the literary Papyrus Anastasi I, and the Papyrus Wilbour from the time of
Ramesses IIIUsimare Ramesses III was the second Pharaoh of the Twentieth Dynasty and is considered to be the last great New Kingdom king to wield any substantial authority over Egypt. He was the son of Setnakhte and Queen TiyMerenese. Ramesses III is believed to have reigned from March 1186 to April 1155 BCE...
records land measurements. In the worker's village of Deir elMedina several
ostracaAn ostracon is a piece of pottery , usually broken off from a vase or other earthenware vessel. In archaeology, ostraca may contain scratchedin words or other forms of writing which may give clues as to the time when the piece was in use...
have been found that record volumes of dirt removed while quarrying the tombs.
Sources
Our understanding of ancient Egyptian mathematics has been impeded by the reported paucity of available sources. The sources we do have include the following texts generally dated to the Middle Kingdom and Second Intermediate Period:
 The Moscow Mathematical Papyrus
The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
 The Egyptian Mathematical Leather Roll
The Egyptian Mathematical Leather Roll was a 10 × 17 in leather roll purchased by Alexander Henry Rhind in 1858...
 The Lahun Mathematical Papyri
The Lahun Mathematical Papyri are part of a collection of Kahun Papyri discovered at ElLahun by Flinders Petrie during excavations of a worker's town near the pyramid of Sesostris II...
 The Berlin papyrus
The Berlin Papyrus 6619, commonly known as the Berlin Papyrus, is an ancient Egyptian papyrus document from the Middle Kingdom. This papyrus was found at the ancient burial ground of Saqqara in the early 19th century CE....
was written around 1300 BC
 The Akhmim Wooden Tablet
The Akhmim wooden tablets or Cairo wooden tablets are two ancient Egyptian wooden writing tablets. They each measure about 18 by 10 inches and are covered with plaster. The tablets are inscribed on both sides. The inscriptions on the first tablet includes a list of servants, which is followed...
.
 The Reisner Papyrus
The Reisner Papyri date to the reign of Senusret I, who was king of Ancient Egypt in the 19th century BCE. The documents were discovered by Dr. G.A. Reisner during excavations in 190104 in Naga edDeir in southern Egypt. A total of four papyrusrolls were found in a wooden coffin in a tomb. * The...
dates to the early Twelfth dynasty of EgyptThe twelfth dynasty of ancient Egypt is often combined with Dynasties XI, XIII and XIV under the group title Middle Kingdom.Rulers:Known rulers of the twelfth dynasty are as follows :...
and was found in Nag elDeir, the ancient town of This.
 The Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
(RMP) dates from the Second Intermediate Period (circa 1650 BC), but its author, AhmesAhmes was an ancient Egyptian scribe who lived during the Second Intermediate Period and the beginning of the Eighteenth Dynasty . He wrote the Rhind Mathematical Papyrus, a work of Ancient Egyptian mathematics that dates to approximately 1650 BC; he is the earliest contributor to mathematics...
, identifies it as a copy of a now lost Middle KingdomThe Middle Kingdom of Egypt is the period in the history of ancient Egypt stretching from the establishment of the Eleventh Dynasty to the end of the Fourteenth Dynasty, between 2055 BC and 1650 BC, although some writers include the Thirteenth and Fourteenth dynasties in the Second Intermediate...
papyrus. The RMP is the largest mathematical text.
From the New Kingdom we have a handful of mathematical texts and inscription related to computations:
 The Papyrus Anastasi I is a literary text from the New Kingdom. It is written as a (fictional) letter written by a scribe named Hori and addressed to a scribe named Amenemope. A segment of the letter describes several mathematical problems.
 Ostracon Senmut 153 is a text written in hieratic.
 Ostracon Turin 57170 is a text written in hieratic.
 Ostraca from Deir elMedina contain computations. Ostracon IFAO 1206 for instance shows the calculations of volumes, presumably related to the quarrying of a tomb.
Numerals
Ancient Egyptian texts could be written in either
hieroglyphEgyptian hieroglyphs were a formal writing system used by the ancient Egyptians that combined logographic and alphabetic elements. Egyptians used cursive hieroglyphs for religious literature on papyrus and wood...
s or in
HieraticHieratic refers to a cursive writing system that was used in the provenance of the pharaohs in Egypt and Nubia that developed alongside the hieroglyphic system, to which it is intimately related...
. In either representation the number system was always given in base 10. The number 1 was depicted by a simple stroke, the number 2 was represented by two strokes, etc.
The numbers 10, 100, 1000, 10,000 and 1,000,000 had their own hieroglyphs. Number 10 is a hobble for cattle, number 100 is represented by a coiled rope, the number 1000 is represented by a lotus flower, the number 10,000 is represented by a finger, the number 100,000 is represented by a frog and a million was represented by a god with his hands raised in adoration.
Hieroglyphics for Egyptian numerals
1  10  100  1000  10,000  100,000  1,000,000 

Z1 
V20\ 
V1 
M12 
D50 
I8 
C11 
Egyptian numerals date back to the
Predynastic periodThe Prehistory of Egypt spans the period of earliest human settlement to the beginning of the Early Dynastic Period of Egypt in ca. 3100 BC, starting with King Menes/Narmer....
. Ivory labels from
AbydosAbydos is one of the most ancient cities of Upper Egypt, and also of the eight Upper Nome, of which it was the capital city. It is located about 11 kilometres west of the Nile at latitude 26° 10' N, near the modern Egyptian towns of el'Araba el Madfuna and alBalyana...
record the use of this number system. It is also common to see the numerals in offering scenes to indicate the number of items offered. The King's Daughter Neferetiabet is shown with an offering of 1000 oxen, bread, beer, etc.
The Egyptian number system was additive. Large numbers were represented by collections of the glyphs and the value was obtained by simply adding the individual numbers together.
The Egyptians almost exclusively used fractions of the form 1/n. One notable exception is the fraction 2/3 which is frequently found in the mathematical texts. Very rarely a special glyph was used to denote 3/4. The fraction 1/2 was represented by a glyph that may have depicted a piece of linen folded in two. The fraction 2/3 was represented by the glyph for a mouth with 2 (different sized) strokes. The rest of the fractions were always represented by a mouth superimposed over a number.
Hieroglyphics for some Egyptian fractions
    

Aa13 
r:Z2 
D22 
r:Z1*Z1*Z1*Z1 
r:Z1*Z1*Z1*Z1*Z1 
Multiplication and division
Egyptian multiplication was done by repeated doubling of the number to be multiplied (the multiplicand), and choosing which of the doublings to add together (essentially a form of
binaryThe binary numeral system, or base2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base2 system is a positional notation with a radix of 2...
arithmetic), a method that links to the Old Kingdom. The multiplicand was written next to the figure 1; the multiplicand was then added to itself, and the result written next to the number 2. The process was continued until the doublings gave a number greater than half of the
multiplierThe term multiplier may refer to:In electrical engineering:* Binary multiplier, a digital circuit to perform rapid multiplication of two numbers in binary representation* Analog multiplier, a device that multiplies two analog signals...
. Then the doubled numbers (1, 2, etc.) would be repeatedly subtracted from the multiplier to select which of the results of the existing calculations should be added together to create the answer.
As a short cut for larger numbers, the multiplicand can also be immediately multiplied by 10, 100, etc.
For example, Problem 69 on the Rhind Papyrus (RMP) provides the following illustration, as if Hieroglyphic symbols were used (rather than the RMP's actual hieratic script).
To multiply 80 × 14 
Egyptian calculation 

Modern calculation 
Result 
Multiplier 

Result 
Multiplier 
V20*V20*V20*V20:V20*V20*V20*V20 
Z1 

80 
1 
V1*V1*V1*V1:V1*V1*V1*V1 
V20 
√ 
800 
10 
V20*V20*V20:V20*V20*V20V1 
Z1*Z1 

160 
2 
V20:V20V1*V1:V1 
Z1*Z1*Z1*Z1 
√ 
320 
4 
V20:V20V1M12 
V20*Z1*Z1*Z1*Z1 

1120 
14 
The √ denotes the intermediate results that are added together to produce the final answer.
The table above can also be used to divide 1120 by 80. We would solve this problem by finding the quotient (80) as the sum of those multipliers of 80 that add up to 1120. In this example that would yield a quotient of 10+4=14. A more complicated example of the division algorithm is provided by Problem 66. A total of 3200 ro of fat are to be distributed evenly over 365 days.
Dividing 3200 by 365
1 
365 

2 
730 

4 
1460 

8 
2920 
√ 
2/3 
243 1/3 
√ 
1/10 
36 1/2 
√ 
1/2190 
1/6 
√ 
First the scribe would double 365 repeatedly until the largest possible multiple of 365 is reached, which is smaller than 3200. In this case 8 times 365 is 2920 and further addition of multiples of 365 would clearly give a value greater than 3200. Next it is noted that
times 365 gives us the value of 280 we need. Hence we find that 3200 divided by 365 must equal
.
Algebra
Egyptian algebra problems appear in both the
Rhind mathematical papyrusThe Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
and the
Moscow mathematical papyrusThe Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
as well as several other sources.
Aha problems involve finding unknown quantities (referred to as Aha) if the sum of the quantity and part(s) of it are given. The
Rhind Mathematical PapyrusThe Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. For instance problem 19 asks one to calculate a quantity taken 1 and ½ times and added to 4 to make 10. In other words, in modern mathematical notation we are asked to solve the
linear equationA linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
:
Solving these Aha problems involves a technique called Method of false position. The technique is also called the method of false assumption. The scribe would substitute an initial guess of the answer into the problem. The solution using the false assumption would be proportional to the actual answer, and the scribe would find the answer by using this ratio.
The mathematical writings show that the scribes used (least) common multiples to turn problems with fractions into problems using integers. The multiplicative factors were often recorded in red ink and are referred to as Red auxiliary numbers.
The use of the Horus eye fractions shows some (rudimentary) knowledge of geometrical progression. Knowledge of arithmetic progressions is also evident from the mathematical sources.
Geometry
We only have a limited number of problems from ancient Egypt that concern geometry. Geometric problems appear in both the
Moscow Mathematical PapyrusThe Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
(MMP) and in the
Rhind Mathematical PapyrusThe Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
(RMP). The examples demonstrate that the Ancient Egyptians knew how to compute areas of several geometric shapes and the volumes of cylinders and pyramids.
 Area:
 Triangles: The scribes record problems computing the area of a triangle (RMP and MMP).
 Rectangles: Problems regarding the area of a rectangular plot of land appear in the RMP and the MMP. A similar problem appears in the Lahun Mathematical Papyri
The Lahun Mathematical Papyri are part of a collection of Kahun Papyri discovered at ElLahun by Flinders Petrie during excavations of a worker's town near the pyramid of Sesostris II...
in London.
 Circles: Problem 48 of the RMP compares the area of a circle (approximated by an octagon) and its circumscribing square. The mathematical machinery employed here resembled the modern concept of a Hilbert Space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
, lacking only in the explicit formalism. This problem's result is used in problem 50, where the scribe finds the area of a round field of diameter 9 khet.
 Hemisphere: Problem 10 in the MMP finds the area of a hemisphere.
 Volumes:
 Cylindrical granaries: Several problems compute the volume of cylindrical granaries (RMP 4143), while problem 60 RMP seems to concern a pillar or a cone instead of a pyramid. It Is rather small and steep, with a seked (slope) of four palms (per cubit). In section IV.3 of the Lahun Mathematical Papyri
The Lahun Mathematical Papyri are part of a collection of Kahun Papyri discovered at ElLahun by Flinders Petrie during excavations of a worker's town near the pyramid of Sesostris II...
the volume of a granary with a circular base is found is using the same procedure as RMP 43.
 Rectangular granaries: Several problems in the Moscow Mathematical Papyrus
The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
(problem 14) and in the Rhind Mathematical PapyrusThe Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
(numbers 44, 45, 46) compute the volume of a rectangular granary.
 Truncated pyramid (frustum): The volume of a truncated pyramid is computed in MMP 14.
The Seqed
Problem 56 of the RMP indicates an understanding of the idea of geometric similarity. This problem discusses the ratio run/rise, also known as the seqed. Such a formula would be needed for building pyramids. In the next problem (Problem 57), the height of a pyramid is calculated from the base length and the seked (Egyptian for slope), while problem 58 gives the length of the base and the height and uses these measurements to compute the seqed. In Problem 59 part 1 computes the seqed, while the second part may be a computation to check the answer: If you construct a pyramid with base side 12 [cubits] and with a seqed of 5 palms 1 finger; what is its altitude?
Further reading
 Boyer, Carl B. 1968. History of Mathematics. John Wiley. Reprint Princeton U. Press (1985).
 Chace, Arnold Buffum. 1927–1929. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. 2 vols. Classics in Mathematics Education 8. Oberlin: Mathematical Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0873531337
 Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0871692325
 Couchoud, Sylvia. 1993. Mathématiques égyptiennes: Recherches sur les connaissances mathématiques de l'Égypte pharaonique. Paris: Éditions Le Léopard d'Or
 Daressy, G. "Ostraca," Cairo Museo des Antiquities Egyptiennes Catalogue General Ostraca hieraques, vol 1901, number 2500125385.
 Gillings, Richard J. 1972. Mathematics in the Time of the Pharaohs. MIT Press. (Dover reprints available).
 Peet, Thomas Eric. 1923. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. London: The University Press of Liverpool limited and Hodder & Stoughton limited
 Robins, R. Gay. 1995. "Mathematics, Astronomy, and Calendars in Pharaonic Egypt". In Civilizations of the Ancient Near East, edited by Jack M. Sasson, John R. Baines, Gary Beckman, and Karen S. Rubinson. Vol. 3 of 4 vols. New York: Charles Schribner's Sons. (Reprinted Peabody: Hendrickson Publishers, 2000). 1799–1813
 Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0714109444
 Sarton, George. 1927. Introduction to the History of Science, Vol 1. Willians & Williams.
 Strudwick, Nigel G., and Ronald J. Leprohon. 2005. Texts from the Pyramid Age. Brill Academic Publishers. ISBN 9004130489.
 Struve, Vasilij Vasil'evič, and Boris Aleksandrovič Turaev. 1930. Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer
 Van der Waerden, B.L. 1961. Science Awakening". Oxford University Press.
 Vymazalova, Hana. 2002. Wooden Tablets from Cairo...., Archiv Orientalni, Vol 1, pages 27–42.
 Wirsching, Armin. 2009. Die Pyramiden von Giza  Mathematik in Stein gebaut. (2 ed) Books on Demand. ISBN 9783837023558.
External links